Least-squares center-to-center and mean object ellipse fabric analysis

The subjectivity of ellipse fitting in many strain techniques has hindered the determination of fabric anisotropy and tectonic strain. However, many sets of x, y co-ordinates can be approximated as an ellipse using a least-squares algorithm to calculate a best-fit ellipse and associated average radial error. For instance, the two dimensional shape of many objects can be approximated as an ellipse by entering digitized co-ordinates of the object margin into the ellipse algorithm.The rim of maximum point density in a normalized Fry diagram is defined by normalized center-to-center distances between touching or nearly touching objects. The enhanced normalized Fry (ENFry) method automates ellipse fitting by entering center-to-center distances between these “touching” objects into the least-squares ellipse algorithm. For homogeneously deformed populations of 200 objects, the ENFry method gives an accurate and precise measure of whole-rock fabric anisotropy, particularly for low ellipticities. When matrix strain exceeds clast strain, manual ellipse fitting of normalized Fry plots gives more accurate matrix anisotropies.The mean object ellipse (MOE) method calculates the best-fit ellipse from the geometry of the objects. Three points from the margin of each object ellipse, centered at the origin and expanded or reduced to unit volume, are used to calculate the best-fit fabric ellipse. The MOE method is very precise for small data sets, making it a good method for mapping heterogenous object strain. However, least-squares calculations maximize the influence of distal and spurious ellipticities, causing the MOE method to overestimate the fabric ellipticity of most aggregates.     

Rapid extraction of central vacancy by image-analysis of Fry plots

The Fry method, based on the relative movement of different material points, typically grain centers, with reference to each other graphically yields a point distribution that displays the finite strain ellipse as a central vacancy. The diffused nature of the central vacancy induces subjectivity in strain estimation, particularly, if the point population when undeformed lacked an isotropic anticlustered distribution. Most existing methods use analytical and/or iterative approaches for improving the sharpness of the central vacancy and positioning the best-fit strain ellipse in a Fry plot. We provide an image-analysis method that is independent of any iteration or analytical solution. It is also an efficient technique for extraction of the central vacancy without any subjectivity. The method is more direct, simple and easy-to-use than most existing techniques.The image-analysis method uses Gaussian blur filter for distinction between the areas of largest and smallest pixel intensities in a Fry plot image. It then applies the optimal threshold value and an inversion filter for extraction of the sharp central vacancy. The method also searches for the best-fit strain ellipse through the extracted central vacancy and displays axial ratio and orientation of the ellipse in a separate window. The validity of the method is tested using several computer-simulated and natural examples.     

在CorelDRAW平台上进行Fry法有限应变测量的新技术

有限应变测量是韧性变形定量估算的主要手段。Fry法有限应变测量以变形颗粒,如变形砾石、鲕粒、近等轴矿物等为应变标志体,地质实践中较容易获得,是岩石有限应变测量的最普遍手段。其传统测量过程极为繁琐,难以满足大范围应变测量的要求,数据精度也较低,限制了Fry法的应用。前人曾根据Fry法原理设计了计算机程序,但其程序相关编程语言如今已基本不能使用。本研究基于目前地质上较普遍使用的制图软件CorelDRAW平台的VBA编程,设计出一个基于CorelDRAW平台的Fry法应变测量宏程序FRY-1。在新图层上将所有变形颗粒的中心点标出后,运行FRY-1宏程序即可快速得到精确的Fry图解,并计算出相对精确的R值。通过实例与“最邻近心对心法”的测量结果进行的对比,证明该方法简便易操作,并可大幅提高有限应变测量的效率和有效减小误差。     

Influence of object concentration on finite strain and effective viscosity contrast: insights from naturally deformed packstones

Deformed conglomerates and ooidal/oncoidal packstones are commonly used to evaluate finite strain in deformed sedimentary successions. In order to obtain a correct estimate of finite strain, it is necessary to consider not only the different behaviour of matrix and objects, but also object concentration. The analysis of two-component rocks characterised by high values of packing commonly results in a substantial underestimate of bulk strain and of viscosity contrast between objects and matrix. In this study, the effects of the volumetric fraction of competent inclusions on both object and bulk measured finite strain, as well as on apparent viscosity contrast, have been investigated in naturally deformed packstones characterised by variable object concentration on the scale of the hand specimen (and hence for homogenous viscosity contrast). Object finite strain has been obtained by Rf/ analysis, whereas the Fry method provides a measure of whole-rock strain that is also a function of inclusion concentration. Therefore, the finite strain measured by the Fry method is better termed effective bulk strain. In order to investigate the role of object concentration, this parameter has been plotted against object and effective bulk strain, and also against viscosity contrast. These diagrams show that: (i) for high values of packing, measured object and effective bulk strain show values that are significantly lower with respect to the calculated maximum value (that would result in the ideal case of no particle interaction and represents therefore the real bulk strain of the samples); (ii) the viscosity contrast shows lower values with respect to the calculated maximum one (that is equal for the three principal sections of the finite strain ellipsoid), and as packing reaches the maximum value, the viscosity contrast approaches a unit value. Empirical equations have also been found that link object concentration with both object and effective bulk finite strain.     

A computer program for the determination of finite strain using fry method

Fry method enables rapid estimate of finite strain from deformed aggregates such as clastic grains, fossil colonies, oolitic or pisolitic aggregates, prophyroblastic minerals or phenocrysts. It has an advantage over the other methods of finite strain analysis in its very quality of enabling rapid estimation with a reasonable degree of accuracy. Details of the software to prepare a plot using Fry method are outlined. This program has an advantage over other computer based programs on the world wide web in its aesthetic getup, small size, user friendliness and a help file.     

Some limitations of the Rf/φ technique of strain analysis

Finite strain determined from R_f/φ analysis of deformed ellipse distributions cannot be separated into compactional and tectonic parts on the basis of population asymmetry about the orientational mean, as was originally suggested. Furthermore, where there is a ductility contrast between objects and matrix, R_f/φ analysis yields an invalid estimate of the strain suffered by the objects.     

The Effect of finite strain and deformation history of HALABAN area,eastern Arabian shield,Saudi Arabia

The present study aims to evaluate a relationship between the mineralogy and structural analysis in the Halaban area and to document the tectonic evolution of Halaban and Al Amar faults. The collected samples were taken from deformed granitiods rocks (such as granite, gneisses and tonalite), metasedimentary, metavolcanic, metagabbro and carbonate rocks are trend to NE-SW with low dip angle in the Halaban area. These samples were 8 from granite, 14 metagabbro, 6 metavolcanics, 5 tonalite, 6 metasedimentary, 10 gneisses and 8 carbonate rocks. Our results are described for the different axial ratios of deformed rocks as the following: XZ sections range from 1.10 to 4.60 in the Fry method and range from 1.70 to 2.71 in the Rf/? method. YZ sections range from 1.10 to 3.34 in the Fry method and range from 1.62 to 2.63 in the Rf/Phi method. In addition, XY sections range from 1 to 3.51 in the Fry method and range from 1 to 1.27 in the Rf/? method for deformed granite rocks, metasedimentry rocks, and metagabbro. The stretch axes for measured samples in the X direction axes (S_X) variety from 1.06 to 2.53 in the Fry method and vary from 1.20 to 1.45 in the Rf/? method. The values of the Y direction axes (S_Y) vary from 0.72 to 1.43 in the Fry method, which indicates contraction and extension in this direction and vary from 1.13 to 1.37 in the Rf/? method which indicates extension in this direction. Furthermore, the Z direction axes (SZ) varies from 0.09 to 0.89 in the Fry method and from 0.52 to 0.71 in the Rf/? method. The stretches axes in the Z direction (SZ) show a vertical shortening about 11% to 91% in the Fry method and show vertical shortening about 29% to 48% in the Rf/? method. The studied rock units are generally affected by brittle-ductile shear zones, which are sub-parallel to parallel NW or NNW trend. It assumed that different rock types of have similar deformation behavior. Based on these results, it is concluded that the finite strain is accumulated during the metamorphism after that was started the deformation by thrusting activity. The contacts between the different rock types were deformed during thrusting under semi-brittle to ductile deformation conditions by simple shear. A component of vertical shortening is also involved causing subhorizontal foliation in the Halaban area.     

Volume strain, strain type and flow path in a narrow shear zone

This study explores the state of finite strain and changes in the mean kinematic vorticity number, grain size, whole-rock chemistry and mineralogy across an upper amphibolite-facies shear zone in a metadiorite, northern Malawi, east-central Africa. P–T conditions during shear-zone formation and deformation were approximately 700–750^?°C and 5–7?kbar and are slightly less than P–T conditions for the regional peak of metamorphism. The major rock-forming minerals, plagioclase, hornblende, biotite, and quartz, were deformed by crystal-plastic processes accompanied by, except for hornblende, dynamic recrystallization. The modal abundance of all four major rock-forming minerals shows no systematic change from the country rock into and across the shear zone, indicating that shear-zone development was not associated with retrograde mineral reactions. The grain size of the major rock-forming minerals decreases within the shear zone. Plagioclase and hornblende, which occur as porphyroblasts outside the shear zone, exhibit a bimodal grain-size distribution within the shear zone. Quartz has a unimodal grain-size distribution in the shear zone. Major and trace element chemistry does not change systematically across the shear zone, implying no volume change in the mylonite. Matrix strain data for plagioclase and hornblende by the Fry method and fabric strain as deduced from R_f/φ analysis of plagioclase and quartz grains demonstrate a slightly constrictional strain type (K≈1.5) across the shear zone. The quantitative finite-strain data for the different residual minerals as obtained by unlike methods show no systematic variation, but recrystallized plagioclase grains record higher strain than the residual grains. The mean kinematic vorticity number changes from approximately 0.3 outside to approximately 0.8 within the shear zone, indicating that the bulk deformation path deviated from progressive simple shear. The estimates for finite strain and the degree of noncoaxiality account for approximately 50% of thinning normal to the shear zone.     

岩质边坡楔形体稳定模糊随机可靠性研究

在统计矩点估计法和模糊随机理论的基础上,提出了楔形体模糊随机可靠性分析改进的模糊点估计法,将楔形体稳定极限状态方程由模糊随机集向普通随机集转化,首先采用Bayes推断方法求解小样本条件下结构面力学参数的概率分布函数,用正态模糊数对随机变量进行模糊随机化处理,然后利用模糊点估计法求解楔形体的可靠度指标。算例分析结果表明,该法使用简便,结果可靠,更能客观地反映楔形体的真实状态,从而为工程决策提供依据。     

均匀离散分布的平面应变过程和Fry法应变测量

本文运用 Apple Macitonsh 计算机对无应变的均匀离散点分布进行系列平面应变模拟,应变叠加模式分别采用具普遍意义的成岩压实+顺层缩短(LPS)+压溶作用和成岩压实+简单剪切+压溶作用。对各应变阶段的变形点分布进行相应的 Fry 法应变测量,并配合低变形砂岩样品的 Fry 法应变分析实例.证实 Fry 法应变测量方法为一非常有用的应变测量方法,其结果不仅能较好地揭示全岩总应变特征,而且能揭示出许多应变叠加的信息,Fry 法揭示的全岩有限应变椭球主面的方位也较为真实可靠。而 Fry 法运用于应变分布不均匀的劈理化岩石中时,能揭示不同变形域的应变特征,从而达到应变分解的目的。     

Strain analysis and microstructural evolution characteristic of neoproterozoic rocks associations of Wadi El Falek,centre Eastern Desert,Egypt

The estimation of finite strain in rocks is fundamental to a meaningful understanding of deformational processes and products on all scales from microscopic fabric development to regional structural analyses. The R_f/φ and Fry methods on feldspar porphyroclasts and mafic grains from 5 granite, 1 metavolcanic, 3 metasedimentary and 1 granodiorite samples were used in Wadi El Falek region. Finite-strain data shows that a high to moderate range of deformation of the granitic to metavolcano-sedimentary samples and axial ratios in the XZ section range from 1.60 to 4.10 for the R_f/φ method and from 2.80 to 4.90 for the Fry method. Furthermore, the short axes are subvertical associated with a subhorizontal foliation. We conclude that finite strain in the deformed granite rocks is of the same order of magnitude as that from metavolcano-sedimentary rocks. Furthermore, contacts formed during intrusion of plutons with some faults in the Wadi El Falek area under brittle to semi-ductile deformation conditions. In this case, finite strain accumulated during superimposed deformation on the already assembled nappe structure. It indicates that the nappe contacts formed during the accumulation of finite strain.     

Pure shear to simple shear-dominated transpression during the Eburnean major D2 deformation,Daléma sedimentary basin,eastern Senegal

Field studies in the Palaeoproterozoïc Daléma basin, Kédougou-Kéniéba Inlier, reveal that the main tectonic feature comprises alternating large shear zones relatively well-separated by weakly deformed surrounding rock domains. Analysis of the various structures in relation to this major D₂ phase of Eburnean deformation indicates partitioning of sinistral transpressive deformation between domains of dominant transcurrent and dominant compressive deformation. Foliation is mostly oblique to subvertical and trending 0–30° N, but locally is subhorizontal in some thrust-motion shear zones. Foliation planes of shear zones contain a superimposed subhorizontal stretching lineation which in places cross-cuts a steeply plunging stretching lineation which is clearly expressed in the metasedimentary rocks of weakly deformed surrounding domains. In the weakly deformed domains, the subhorizontal lineation is absent, whereas the oblique to subvertical lineation is more fully developed. Finite strain analyses of samples from surrounding both weakly deformed and shearing domains, using finite strain ratio and the Fry method, indicate flattened ellipsoid fabrics. However, the orientation of the long axis (X) of the finite strain ellipsoid is horizontal in the shear zones and oblique within the weakly deformed domains. Exceptionally, samples from some thrust zones indicate a finite strain ellipsoid in triaxial constriction fabrics with a subhorizontal long axis (X). In addition, the analysis of the strain orientation starting from semi-ductile and brittle structures indicates that a WNE–ESE (130° N to 110° N) orientation of strain shortening axis occurred during the Eburnean D₂ deformation.     

Effect of grain packing tightness on strain estimation from the Fry method

How tightly should a sample be packed for strain estimation by the Fry method? We address this issue using synthetic simulations of 900 images such that each image contains 200 randomly distributed grains, but differs from other images with respect to the packing tightness and the degree of sorting. Each image is coaxially distorted by various known strain ratios and the strain estimates from distorted images are obtained by the Fry method. The statistical errors in the strain estimates are found to grow larger with the decrease in the packing tightness irrespective of the level of distortion. We demonstrate that a progressive decrease in the packing tightness results in an increasingly clustered nature of grain center distribution and hence the larger errors. These results, obtained from the synthetic images, are corroborated by two natural examples of sandstone, one loosely packed and the other tightly packed. Based on the results of tests on synthetic and natural examples, we recommend that the Fry method should be used only on those samples that have \({>}30\%\) packing density.     

Strain analysis using deformed line distributions

This paper describes a new technique of strain analysis using passively deformed linear markers which were initially uniformly or symmetrically distributed in a plane. The axial ratio of the strain ellipse is determined from the quartiles or other percentiles of the deformed distribution. The technique permits statistical testing of assumptions and rapid reconstruction of the undeformed state.     

被动椭圆形标志物应变分析的计算机模拟

本文采用计算机数学模拟的方法更深入地探讨了椭圆形标志物应变分析技术(R_f/Φ去应变法、Shimamoto等提出的代数法、调和平均值法和算术平均值法)的适用条件。一共形成了十一组有代表性的应变前数组,四种变形路径:纯剪、单剪、纯剪加单剪和体变。当应变前数组在(-90°,90°)内均匀分布或随机分布时,无论变形路径是哪一种(共轴的还是非共轴的),上述的所有方法都是适用有,且以代数法最佳。当应变前数组具有优选方位(单峰分布)且变形路径为共轴递进变形时,在一定条件下,R_f/Φ法能用来估计应变。经分析发现当应变足够大时,调和平均值法是一种简单快速的应变计算法。     

假想变位法模拟构造应力场的理论与实践

把变形地质体作为弹塑性体,用增量理论计算变形地质体内应力和应变的分布是假想变位法的理论基础。用假想变位法模拟构造形成时变形地质体的物理边界条件,代替历来人为假设物理边界条件的方法,尽量减少人为因素,提高了模拟实验结果的可信性。 把这一方法应用到日本佐渡金矿控矿断裂形成机制的研究中,对控矿断裂成因的地质推断得到了理论验证。     

基于Hoek-Brown经验公式的岩体稳定性可靠度分析

为消除因用Hoek-Brown经验公式的中间变量而导致评估的稳定性概率产生累积误差,提出岩体稳定性可靠度应直接由基本变量入手来评估.简要介绍Hoek-Brown经验公式评估力学参数方法后,根据微元体应力定义了单元安全系数,建立了包含Hoek-Brown经验公式基本变量在内的功能函数.选用了Rosenbleuth点估计法评估单元安全系数的概率特征值,介绍了与有限元法结合计算稳定性可靠度的基本流程.对一开挖边坡稳定性可靠度计算分析表明,计算结果与边坡实际发生的破坏特征一致.建议的方法可方便地用于边坡、地基和地下洞室中岩体稳定性的可靠度评价.     

Comparison of Fry strain ellipse and AMS ellipsoid trends to tectonic fabric trends in very low-strain sandstone of the Appalachian fold–thrust belt

In moderately to highly strained sandstones, both the long axis of the bedding-parallel finite-strain ellipse, as calculated by the normalized Fry method, and the projection of the long axis of the AMS ellipsoid on the plane of bedding, align well with local “structural grain” (trends of cleavage, folds, and faults). This relationship implies that results of both 2D Fry and AMS analyses represent the local layer-parallel tectonic strain component. Do both methods provide comparable results for very low-strain sandstone (e.g., <5%)? To address this question, Fry and AMS analyses were conducted in very low-strain sandstone from two localities in the Appalachian foreland fold–thrust belt: near Rosendale in New York and the Lackawanna synclinorium of Pennsylvania. We compared the map projections of both bedding-parallel Fry ellipses and AMS ellipsoids to the local structural grain. In both study areas, projections of the long axis of Fry strain ellipses do not cluster in a direction parallel to structural grain, whereas the projection of the long axes of AMS ellipsoids do cluster closely to structural grain. This observation implies that in very low-strain sandstone, AMS analysis provides a more sensitive “quick” indicator of tectonic fabric than does normalized Fry analysis.     

用线状被动标志物向分布估测岩石总体应变的最优化方法

在一些假设下提出了利用线状被动标志物的方向数据估测岩石总体应变的最优化方法。该方法等同地考虑到每个测量数据对应变估计的贡献,从而能够较好地反映出岩石的总体应变。它具有较好的稳健和广泛的适用对象。实验表明,观察数据大约在80个或更少时估测的应变椭圆轴比的相对误差小于20％,超过160个观察数据一般小于5％－10％。